Module for Taylor's Method for solving O.D.E.'s

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Taylor's Method of Order 4.  To approximate the solution of the initial value problem    with    over    by evaluating ,  ,    and    and using the Taylor polynomial at each step.

Background. For an arbitrary  ,  what are the derivatives    for   ?

Solution. Right click this cell to view the Mathematica solution.

Animations (Taylor's method of order 3  Taylor's method of order 3).  Internet hyperlinks to animations.

Animations (Taylor's method of order 4  Taylor's method of order 4)s.  Internet hyperlink to animations.

We use the following subroutine TaylorMeth to perform the computations. It starts with the initial point    and generates the sequence of values  ,  where d_k is    evaluated at  .  

Mathematica Subroutine (Taylor's method for D.E.'s).

Example 1.  Use the Taylor method of order    to compute numerical solutions for the differential equation    with initial condition    over the interval  .
Solution  1.

Example 2.  Use Mathematica's procedure DSolve to get an analytic solution for the differential equation    with initial condition    over the interval  .
Solution  2.

Example 3.  Determine the error in the Taylor Series method, by comparing it with the analytic solution.
Solution  3.

Example 4.  Error analysis.  The Taylor series method of order    allegedly has a Final Global Error (FGE) of order  .  Hence, the error at the right endpoint should appear to decrease by when the number of sub-intervals is doubled. Use the D.E. in exercise 1 and investigate this behavior for    sub-intervals of  .  
Solution  4.

Old Lab Project (Taylor's Method for solving ODE's  Taylor's Method for solving ODE's).  
Internet hyperlinks  to an old lab project.  

Research Experience for Undergraduates

Taylor Series Method for ODE's  Taylor Series Method for ODE's  
Internet hyperlinks to web sites and a bibliography of articles.  

Downloads (Taylor Series Method for ODE's Taylor Series Method for ODE's).  

Download this Mathematica notebook.  

(c) John H. Mathews 2003